5 n P j j (P i,, P k, j 1) 1 n n ) φ(n) = n (1 1Pj [ ] φ φ P j j P j j = = = = = n = φ(p j j ) (P j j P j 1 j ) P j j ( 1 1 P j ) P j j ) (1 1Pj (1 1P
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1 p P 1 n n n 1 φ(n) φ φ(1) = 1 1 n φ(n), n φ(n) = φ()φ(n) [ ] n 1 n 1 1 n 1 φ(n) φ() φ(n) φ(abc ) = φ(a)φ(b)φ(c) a, b, c, [ ] 3 p n p n 1 [ ] p n px x 1 x p n 1 x px p p n 1 4 φ(p n ) = p n p n 1 [ ] 3 1/1
2 5 n P j j (P i,, P k, j 1) 1 n n ) φ(n) = n (1 1Pj [ ] φ φ P j j P j j = = = = = n = φ(p j j ) (P j j P j 1 j ) P j j ( 1 1 P j ) P j j ) (1 1Pj (1 1Pj ) P j 1 j (P j 1) 1 φ(009) [ ] φ(009) = φ(7 41) = 7(7 1)(41 1) = y = 0, x = 6, y = x [ ] (x,y) y x x y > 0 y φ(x) x 1 + φ(1) + φ() + φ(3) + φ(4) + φ(5) + φ(6) = =13 /1
3 1 0 1 (0, 0) (1, 0) φ(1) 1 (0, 0) (1, 1) 1 y 6 O 6 x 6 [ ] p q 1 3 p + 1 (1) q p () q q p q q p. 3 (1) () li φ() = a = φ() a 1, a, a 3, ( 1) [ ] 3/1
4 (1) p() φ() p() 6 φ() () 5 φ() li p() = n = ) (1 1Pj p j j 1 φ(p) p = 1 1 p 1 1 a, b a b n n (congruent odulo n) a b ( od n ) 7 a n b, c n b c ab ac [ ] b c n a(b c) n a n 8 n a [ ] n n a φ(n) 1 ( od n ) n a r 1, r, r 3,, r φ(n) (1) ar 1, ar, ar 3,, ar φ(n) () 4/1
5 n i j = ar i ar j ar i ar j a(r i r j ) 0 r i r j n r i r j () (1) a φ(n) r 1 r r 3 r φ(n) r 1 r r 3 r φ(n) ( od n ) a φ(n) 1 9 p a a p 1 1 ( od p ) a [ ] φ(1000) = φ( ) = 5 ( 1)(5 1) = ( od 1000) (81 ) n 1 p = 7, a = ( od 7) 4 3 = 64 1 ( od 7) p 1 p a 1 p 1 5/1
6 n = 1 = 3 7 a φ(1) = a 1 1 ( od 1) a φ(3) = a 1 ( od 3) a φ(7) = a 6 1 ( od 7) (a ) a 6 1 ( od 1) n P j j (P i,, P k, j 1) n [ ] ( j + 1) 11 n P j j (P i,, P k, j 1) n j P i j = i=0 P j+1 j 1 P j 1 P j+1 j 1 (P j 1)( j + 1) [ ] [ ] ( )(1 + 41) = 57 4 = = 399 6/1
7 1 n n nφ(n) n n n n {n φ(n) + 1} n {n φ(n) + 1} {n φ(n)} [ ] n n 0 n = n n n φ(n) + 1 n {n φ(n) + 1} n {n φ(n) + 1} {n φ(n)} n(n + 1) n n nφ(n) n 7/1
8 0 0 n n n 0 n 6 [ ] {009 φ(009) + 1} {009 φ(009)} = 009 { } { } = n n S n 3 S 0 ( od n) [ ] 1 S = nφ(n) 5 n 3 φ(n) p (p 1) φ(n) S n 14 n a, b n = ab a φ(b)+1 a ( od n) b = p p [ ] a p a ( od n) a φ(b)+1 a a(a φ(b) 1) ( od n) a φ(b) 1 b a(a φ(b) 1) 0 ( od n) a n n φ(b) a φ(b) 1 0 (od n) a ak b (ak) φ(b)+1 (ak) ( od n) /1
9 [ ] 1 od = = ,, 4, 5, 8, 10, 11, 13, 16, 17, 19, 0 φ(1) = 1 ( 5 φ(3 7) = (3 1)(7 1) = 1 ) 1 a a = a a 009 a 5 a 1,, 4, 5, 8, 10, 11, 13, 16, 17, 19, 0 (3) a 5 1, 11, 16, 17, 8, 19,, 13, 4, 5, 10, 0 (4) (3) a = (5) a a a a a a a a 009 (4) a a (6) ( ) 9/1
10 (4) 3 5 ( ) 1( ) (5) ( ) 1( ) (7) (6),(7) [ ] 1 1 a 7 a 3 a 7 a 7 a 3 a 1 a 0 1,,6 6 a a 7 a /1
11 ( ) 5 = ( ) 5 (18 14) = 0 15 n a ax b ( od n) (8) x a φ(n) 1 b ( od n) [ ] (8) ax ny = b φ(n) a φ(n) 1 n 16 n n φ(p i ) = p n i=1 [ ] 4 = p n p n 1 + p n 1 p n + + p = p n 17 n d n φ(d) = n [ ] n = P j j (P i,, P k, j 1) 11/1
12 d n φ(d) =(φ(p 1 1 ) + φ(p ) + + φ(1))(φ(p ) + φ(p 1 ) + + φ(1)) (φ(p k k ) + + φ(1)) =P 1 1 P P k k =n 18 n d n 1,, 3,, n (x, d) = d x ( n ) φ d [ ] x = dx, n = dn x 1,, 3,, n n (x, n ) = 1 x φ(n ) 1/1
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